Optimal Estimation of Generic Dynamics by Path-Dependent Neural Jump ODEs

This paper studies the problem of forecasting general stochastic processes using an extension of the Neural Jump ODE (NJ-ODE) framework. While NJ-ODE was the first framework to establish convergence guarantees for the prediction of irregularly observed time series, these results were limited to data stemming from It\^o-diffusions with complete observations, in particular Markov processes where all coordinates are observed simultaneously. In this work, we generalise these results to generic, possibly non-Markovian or discontinuous, stochastic processes with incomplete observations, by utilising the reconstruction properties of the signature transform. These theoretical results are supported by empirical studies, where it is shown that the path-dependent NJ-ODE outperforms the original NJ-ODE framework in the case of non-Markovian data. Moreover, we show that PD-NJ-ODE can be applied successfully to limit order book (LOB) data

Estimating Full Lipschitz Constants of Deep Neural Networks

We estimate the Lipschitz constants of the gradient of a deep neural network and the network itself with respect to the full set of parameters. We first develop estimates for a deep feed-forward densely connected network and then, in a more general framework, for all neural networks that can be represented as solutions of controlled ordinary differential equations, where time appears as continuous depth. These estimates can be used to set the step size of stochastic gradient descent methods, which is illustrated for one example method

Extending Path-Dependent NJ-ODEs to Noisy Observations and a Dependent Observation Framework

The Path-Dependent Neural Jump ODE (PD-NJ-ODE) is a model for predicting continuous-time stochastic processes with irregular and incomplete observations. In particular, the method learns optimal forecasts given irregularly sampled time series of incomplete past observations. So far the process itself and the coordinate-wise observation times were assumed to be independent and observations were assumed to be noiseless. In this work we discuss two extensions to lift these restrictions and provide theoretical guarantees as well as empirical examples for them

Optimal Stopping via Randomized Neural Networks

This paper presents the benefits of using randomized neural networks instead of standard basis functions or deep neural networks to approximate the solutions of optimal stopping problems. The key idea is to use neural networks, where the parameters of the hidden layers are generated randomly and only the last layer is trained, in order to approximate the continuation value. Our approaches are applicable to high dimensional problems where the existing approaches become increasingly impractical. In addition, since our approaches can be optimized using simple linear regression, they are easy to implement and theoretical guarantees can be provided. We test our approaches for American option pricing on Black--Scholes, Heston and rough Heston models and for optimally stopping a fractional Brownian motion. In all cases, our algorithms outperform the state-of-the-art and other relevant machine learning approaches in terms of computation time while achieving comparable results. Moreover, we show that they can also be used to efficiently compute Greeks of American options

Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricing

We propose an optimal iterative scheme for federated transfer learning, where a central planner has access to datasets ${\cal D}_1,\dots,{\cal D}_N$ for the same learning model $f_{\theta}$. Our objective is to minimize the cumulative deviation of the generated parameters $\{\theta_i(t)\}_{t=0}^T$ across all $T$ iterations from the specialized parameters $\theta^\star_{1},\ldots,\theta^\star_N$ obtained for each dataset, while respecting the loss function for the model $f_{\theta(T)}$ produced by the algorithm upon halting. We only allow for continual communication between each of the specialized models (nodes/agents) and the central planner (server), at each iteration (round). For the case where the model $f_{\theta}$ is a finite-rank kernel regression, we derive explicit updates for the regret-optimal algorithm. By leveraging symmetries within the regret-optimal algorithm, we further develop a nearly regret-optimal heuristic that runs with $\mathcal{O}(Np^2)$ fewer elementary operations, where $p$ is the dimension of the parameter space. Additionally, we investigate the adversarial robustness of the regret-optimal algorithm showing that an adversary which perturbs $q$ training pairs by at-most $\varepsilon>0$, across all training sets, cannot reduce the regret-optimal algorithm's regret by more than $\mathcal{O}(\varepsilon q \bar{N}^{1/2})$, where $\bar{N}$ is the aggregate number of training pairs. To validate our theoretical findings, we conduct numerical experiments in the context of American option pricing, utilizing a randomly generated finite-rank kernel.Comment: 54 pages, 3 figure

Denise: Deep Learning based Robust PCA for Positive Semidefinite Matrices

The robust PCA of high-dimensional matrices plays an essential role when isolating key explanatory features. The currently available methods for performing such a low-rank plus sparse decomposition are matrix specific, meaning, the algorithm must re-run each time a new matrix should be decomposed. Since these algorithms are computationally expensive, it is preferable to learn and store a function that instantaneously performs this decomposition when evaluated. Therefore, we introduce Denise, a deep learning-based algorithm for robust PCA of symmetric positive semidefinite matrices, which learns precisely such a function. Theoretical guarantees that Denise's architecture can approximate the decomposition function, to arbitrary precision and with arbitrarily high probability, are obtained. The training scheme is also shown to convergence to a stationary point of the robust PCA's loss-function. We train Denise on a randomly generated dataset, and evaluate the performance of the DNN on synthetic and real-world covariance matrices. Denise achieves comparable results to several state-of-the-art algorithms in terms of decomposition quality, but as only one evaluation of the learned DNN is needed, Denise outperforms all existing algorithms in terms of computation time

Transcriptome-pathology correlation identifies interplay between TDP-43 and the expression of its kinase CK1E in sporadic ALS.

Sporadic amyotrophic lateral sclerosis (sALS) is the most common form of ALS, however, the molecular mechanisms underlying cellular damage and motor neuron degeneration remain elusive. To identify molecular signatures of sALS we performed genome-wide expression profiling in laser capture microdissection-enriched surviving motor neurons (MNs) from lumbar spinal cords of sALS patients with rostral onset and caudal progression. After correcting for immunological background, we discover a highly specific gene expression signature for sALS that is associated with phosphorylated TDP-43 (pTDP-43) pathology. Transcriptome-pathology correlation identified casein kinase 1ε (CSNK1E) mRNA as tightly correlated to levels of pTDP-43 in sALS patients. Enhanced crosslinking and immunoprecipitation in human sALS patient- and healthy control-derived frontal cortex, revealed that TDP-43 binds directly to and regulates the expression of CSNK1E mRNA. Additionally, we were able to show that pTDP-43 itself binds RNA. CK1E, the protein product of CSNK1E, in turn interacts with TDP-43 and promotes cytoplasmic accumulation of pTDP-43 in human stem-cell-derived MNs. Pathological TDP-43 phosphorylation is therefore, reciprocally regulated by CK1E activity and TDP-43 RNA binding. Our framework of transcriptome-pathology correlations identifies candidate genes with relevance to novel mechanisms of neurodegeneration

A switch for epitaxial graphene electronics: Utilizing the silicon carbide substrate as transistor channel

Due to the lack of graphene transistors with large on/off ratio, we propose a concept employing both epitaxial graphene and its underlying substrate silicon carbide (SiC) as electronic materials. We demonstrate a simple, robust, and scalable transistor, in which graphene serves as electrodes and SiC as a semiconducting channel. The common interface has to be chosen such that it provides favorable charge injection. The insulator and gate functionality is realized by an ionic liquid gate for convenience but could be taken over by a solid gate stack. On/off ratios exceeding 44000 at room temperature are found

Axon-Specific Mitochondrial Pathology in SPG11 Alpha Motor Neurons

Pathogenic variants in SPG11 are the most frequent cause of autosomal recessive complicated hereditary spastic paraplegia (HSP). In addition to spastic paraplegia caused by corticospinal degeneration, most patients are significantly affected by progressive weakness and muscle wasting due to alpha motor neuron (MN) degeneration. Mitochondria play a crucial role in neuronal health, and mitochondrial deficits were reported in other types of HSPs. To investigate whether mitochondrial pathology is present in SPG11, we differentiated MNs from induced pluripotent stem cells derived from SPG11 patients and controls. MN derived from human embryonic stem cells and an isogenic SPG11 knockout line were also included in the study. Morphological analysis of mitochondria in the MN soma versus neurites revealed specific alterations of mitochondrial morphology within SPG11 neurites, but not within the soma. In addition, impaired mitochondrial membrane potential was indicative of mitochondrial dysfunction. Moreover, we reveal neuritic aggregates further supporting neurite pathology in SPG11. Correspondingly, using a microfluidic-based MN culture system, we demonstrate that axonal mitochondrial transport was significantly impaired in SPG11. Overall, our data demonstrate that alterations in morphology, function, and transport of mitochondria are an important feature of axonal dysfunction in SPG11 MNs